The heptagon is a polygon that has seven sides and seven interior angles. The sum of all the interior angles of a heptagon is equal to 900°. Depending on the characteristics we take, we can distinguish different types of heptagons. For example, if we consider the length of the sides and the measure of the angles, we can identify regular and irregular heptagons. If we consider the contour, we can identify convex and concave heptagons.

Here, we will look at a brief definition of heptagons and we will learn about some types of heptagons. In addition, we will discover some of the fundamental characteristics of these geometric figures. Finally, we will learn the most important formulas for regular heptagons and use them to solve some problems.

## Definition of a heptagon

We can define a heptagon as a polygon with seven sides and seven interior angles. Recall that, a polygon is a two-dimensional closed figure made up of straight segments. The heptagon is formed by all its sides, which meet each other end to end. Depending on different characteristics, we can distinguish the following types of heptagons:

- Regular and irregular
- Convex and concave

### Irregular and regular heptagons

Regular heptagons have the characteristic of having sides of the same length and angles with the same measure. Irregular heptagons have the characteristic of having sides of different lengths, angles of different measures, or both at the same time.

### Convex and concave heptagons

Convex heptagons are polygons that have all their vertices pointing outward. Concave heptagons are polygons that have at least one vertex that is pointing inward.

## Fundamental characteristics of a heptagon

Heptagons have the following fundamental characteristics:

- In a heptagon, the sum of all the interior angles is 900°.
- A regular heptagon has seven sides of equal length.
- A regular heptagon has seven angles with the same measure.
- The total sum of the interior angles is 360°
- Each internal angle of a regular heptagon measures 128.57°.
- The total number of diagonals in a regular heptagon is 14.
- The number of triangles formed in a heptagon is seven.

## Important heptagon formulas

The most important formulas for a heptagon are the perimeter, area, and apothem formulas.

### Formula for the perimeter of a regular heptagon

The perimeter is the sum of the lengths of all the sides of a heptagon. Since a regular heptagon has seven equal sides, we have:

$latex p=7s$ |

where *s* is the length of one of the sides of the hexagon.

### Formula for the area of a regular heptagon

The formula for the area of a heptagon is:

$latex A= \frac{7}{2}sa$ |

where, *s* is the length of one of the sides and *a* is the length of the apothem.

### Formula of the apothem of a regular heptagon

We can find the apothem of a heptagon using the following formula:

$latex a= \frac{s}{2tan(25.71°)}$ |

where, *s* is the length of one side of the hexagon.

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## Examples of heptagon problems

### EXAMPLE 1

- What is the perimeter of a heptagon with sides of length 13 m?

**Solution:** We use $latex s=13$ in the formula for the perimeter:

$latex p=7s$

$latex p=7(13)$

$latex p=91$

The perimeter is 91 m.

### EXAMPLE 2

- What is the area of a heptagon that has sides of length 11 m and an apothem of 11.42 m?

**Solution:** We have $latex s=11$ and $latex a=11.42$. Therefore, we use these lengths in the formula for the area:

$latex A= \frac{7}{2}sa$

$latex A= \frac{7}{2}(11)(11.42)$

$latex A=439.67$

The area of the heptagon is 439.67 m².

### EXAMPLE 3

- What is the apothem of a heptagon that has sides of length 10 m?

**Solution:** We use the apothem formula with $latex s=10$:

$latex a= \frac{s}{2\tan(25.71)}$

$latex a= \frac{10}{2\tan(25.71)}$

$latex a=10.38$

The length of the apothem is 10.38 m.

## Heptagon – Practice problems

## See also

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